entropy of a Fermi dirac ideal gas

1. The problem statement, all variables and given/known data
From the expression of the partition function of a fermi dirac ideal gas
ln(Z)=αN + ∑ ln(1+exp(-α-βEr))
show that
S= k ∑ [ nrln(nr)+(1-<nr>)ln(1-<nr>)

2. Relevant equations
S=k( lnZ+β<E>)
<nr>=-1/β ∂ln(Z)/∂Er

3. The attempt at a solution
I tried to start with
S=k( lnZ+β<E>)
But I don’t know how we get to introduce the nr in that ?



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